A STUDY ON THE NONLINEALITY OF RUNOFF PHENOMENA
AND ESTIMATION OF EFFECTIVE RAINFALL
SHUICHI KURE
Graduate school, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551 Japan
TADASHI YAMADA
Dept. of Civil Engineering, Faculty of Science and Engineering, Chuo University, 1-1327 Kasuga, Bunkyo-ku, Tokyo, 112-8551 Japan
The purpose of this study is to clarify the nonlinearity of runoff phenomena and to
understand hydraulic processes in mountainous basins. A Universal lumped analysis
method for runoff in a mountainous slope is proposed. The partial differential equation of
kinematic wave theory applied to the subsurface flow is transformed to an ordinary
differential equation with respect to the discharge along the slope by separation of
variables under the assumption that the slope-length is short and the thickness of surface
soil layer is small. Runoff parameters are determined in terms of the slope gradient, slope
length, thickness of surface soil layer, unsaturated hydraulic conductivity and effective
porosity in the model. This model can express the characteristics of nonlinearity of runoff
and base-flow recession on small mountainous basins. Two new methods to estimate
effective rainfall are proposed. One is to estimate effective rainfall from observed data of
discharge. Another method is to estimate effective rainfall by theory of water holding
capacity of a basin. We compared two methods of estimating effective rainfall and
applied the methods to estimate effective rainfall to the Kusaki dam basin in Japan. The
results of runoff analysis match well with the observed data in Kusaki dam basin. It can
be concluded that the proposed lumped analysis method can express the runoff in a
mountainous slope effectively.
INTRODUCTION
The method of the runoff analysis as that of rational function, storage function, tank
model and kinematic wave method have been researched for a flood forecasting. The
conditions of a practical runoff model are showed as follows. First of all, it is easy to
simulate a runoff in the mountainous basins. Secondary, the parameter of runoff model
can be determined in the mountainous basins that don’t have the past hydrological data.
Those methods aren’t filled with the former conditions.
The purpose of the present paper is to construct the practical and physical flood
forecasting system for the mountainous basins. In addition, this system expressed by
runoff parameters can easily apply to the runoff analysis in any topography and
morphology as to river, mountainous and urban basins. First, Interflow is the main
component of direct runoff in a mountainous slope. We assume it to be the overland flow.
The partial differential equation of kinematic wave theory applied to the subsurface flow
1
2
is transformed to an ordinary differential equation with respect to the discharge along the
slope by separation of variables the method of storage function by the mountainous slope
lumped. Next, runoff parameters are determined by morphological or geological
quantities with using the unsaturated theory. Finally, we estimate effective rainfall from
observed data of discharge.
LUMPING MODEL FOR RUNOFF ANALYSIS IN A SLOPE
Lumping of subsurface flow
We make the runoff model of the interflow. This flow is the main component of direct
runoff in a mountainous slope. Eq.(1) is a motion rule generalized for each slope. The
conservation of mass equation in a mountainous slope can be obtain as Eq.(2),
v  h m , q  vh  h m  1
h q

 r (t )
t x
(1)
(2)
where h(mm) is the ponding depth in the surface soil layer, v(mm/h) is the surface flow
velocity, r(mm/h) is the effective rainfall and q is the discharge per unit length along
bottom of the slope. From Eq.(1) and Eq.(2), the expression for the q can be obtain as
follows,
q
q
 aq m 1
 aq m 1 r t 
t
x
m
m
(3)
1
where
a  m  1 m1
The direct runoff is composed by the discharge from the partial source area. This
area is channel and moisture soil layer near that. Then, we can suppose that the slope
length is short sufficiently. In this case, we can consider that the longer slope length, the
more discharge increases. We can transform q(x, t) to Eq.(4) as the approximate equation
and the separation of variables, Eq.(3) is lumped as follows that is the ordinary
differential equation,
qx, t   x  q t 
dq
 a0 q (r (t )  q )
dt
(4)
(5)
1
1
where a0  aL 1  (m  1) m 1 Lm 1 ,   m
m 1
where q* is runoff rate (mm/h), x is optional coordinate (m) on this slope. Eq.(5) becomes
a fundamental equation which shows runoff from a slope.
3
The relationship between unsaturated theory and kinematic wave equation
Because of the runoff parameters have not been determined by morphological quantities
until the former, we investigate relationship between unsaturated flow equation and
kinematic wave equation in this section. We express that relationship between the lumped
model expressed in this paper and the unsaturated flow model, that is proposed by Suzuki
and Kubota, in the surface soil layer as follows.
The unsaturated flow is expressed by Richards equation as follows,

   
   



 sin   
 cos  
k 
k 
t
x   x
 z   z

c
(6)
where Ψ is a pressure head, ω is a slope gradient, c is the specific moisture capacity, θ
is the volumetric soil moisture and k is the unsaturated hydraulic conductivity. Suzuki
reported that the recession of discharge is not almost influenced by the fist and second
terms of Eq.(6) neglected on the right-hand when the surface soil layer thickness is thin.
In this case, we have Eq.(7) as follows,
c

  
k


  sin 
t
 t
t
x
(7)
We integrate the Eq.(7) with respect to z-directions. This integrates means that the
volumetric soil moisture and the unsaturated hydraulic conductivity is averaged with
respected to z-directions. We have Eq.(8) as follows,
D

k
  D sin 
 r (t )
t
x
(8)
where D is the thickness of surface soil layer and r(t) is the effective rainfall. Eq.(8)
transformed from the function with respect to the z-directions to the function with respect
to the x-directions (slope-directions). Eq.(8) is kinematic wave equation. We use Kozney
equation (9) as the relationship between the effective saturation Se and the unsaturated
hydraulic conductivity k. We use Eq.(10) as the relationship between the volumetric soil
moisture θand the effective porosity w. These quantities is respectively averaged with
respect to z-directions as follows,
k  k s S e
  Se  w  r

(9)
(10)
k  ks Se
(11)
  Se  w r
(12)
Inserting Eq.(11) and Eq.(12) in Eq.(8) yields
4
Dw
S e
S 
  Dk s sin  e  r (t )
t
x
(13)
where ks is the saturated hydraulic conductivity, γ is exponent in hydraulic
conductivity function and w is the effective porosity. We expand the Darcy’s law into
unsaturated zone and average it with respect to z-directions as follows,
vk


 k s S e
 k s S e
z
z
(14)
where v is the infiltration velocity of vertical direction. Eq.(14) is transformed with
respect to the discharge per unit length q. It is given by Kubota as Eq.(15). We transform
Eq.(15) as Eq.(16). Inserting Eq.(16) in Eq.(13) and transformed with respect to the
discharge per unit length q yields
q  k s DS e sin 
(15)
1


q

S e ( x, t )  
 k s D sin  
D
 1

(16)
1
wk s

1

sin 

1

q  q

 r (t )
t
x
(17)
On the other hand, inserting Eq. (1) in Eq. (2) yields
1
1
 1  m 1 q m 1 q

 r (t )
 
t
x
 
(18)
Eq.(17) and Eq.(18) is the equivalent equations, because the two point of view have
emerged over the one phenomenon of the subsurface flow in the mountainous slope .
Comparing Eq.(17) with Eq.(18) yields
  m 1
k sin 
  s  1 
D w
(19)
(20)
It is important that runoff parameters that have been calibrated by the actual
discharge can be uniquely expressed by quantities of a surface soil layer, a saturated
hydraulic conductivity, an exponent in hydraulic conductivity function, the surface soil
layer thickness and the effective porosity.
5
THE ANALYTICAL SOLUTION WHICH SHOWS RUNOFF
Analytical solution related to recession curve of hydrograph
Eq. (5) which is fundamental equation of runoff has analytical solution related to
recession curve of hydrograph.
To express recession curve of hydrograph, the origin of time:t is point of time in
which the rainfall is stopped and r(t)=0. Eq.(21) as a separation of variables is obtained
from Eq.(5). We can be obtain analytical solution related to recession curve of
hydrograph from Eq.(21) on q**(0)=q**0 and β≠0. It is expressed by Eq.(22).
1
dq t 
 a0
 1
dt
q t 
q 0
q t  
1  a0 q0t 1 / 
(21)
(22)
where q**:runoff rate[mm/h] in recession curve. Eq.(22) is a peculiar solution of
nonlinear equation. If condition is on β=0, which means linear condition, we obtain
analytical solution Eq.(23) as the exponential function. We know recession curve of
hydrograph is diminished as the fraction function from observed data of discharge. We
could say recession curve of hydrograph diminished nonlinearly.
If value of β which value takes from 0 to 1.0 is 0.5 because of lumping of runoff
process, we can be obtain analytical solution Eq.(24).
q t  
q0
1  1/ 2a
0
q0  t

2
q t   q 0 exp  a0t 
(23)
(24)
Eq.(23) and Eq.(24) are completely same with analytical solution which obtained
Werner and Sunquist, Roche and Takagi. It is very interesting that same analytical
solution is obtained from the equations applied in different flow field.
Analytical solution related to hydrograph
Eq.(5) as fundamental equation which shows runoff phenomena is nonlinear equation.
Generally, nonlinear equations can obtain analytical solution under peculiar conditions.
Eq.(5) can obtain analytical solution under condition of β=1. On β=1, Eq.(5) is
transformed Eq.(25) which is the equation of Bernoulli type as follows, On q*(0)=q*i, we
can be obtain analytical solution Eq.(26) as follows,
dq* t 
2
 a 0 r (t )q* t   a 0 q t   0
dt
(25)
a 0 r t dt
t
q* t  
q*i e 0
t a 0 r  d
q*i a0  e 0
dt  1
t
0
(26)
6
In analytical solution Eq.(26), Integrated value of the rainfall goes into exponential
function and the effect of the rainfall has been expressed nonlinearly. Initial condition is
expressed in fraction function form not simple function form. The nonlinear effect of
initial condition is apparently expressed in analytical solution.
NONLINEAR CHARACTERISTICS IN THE RUNOFF PHENOMENA
It is said that runoff phenomena from the mountainous basins shows the nonlinearity
generally. The example of examining the factor which shows the nonlinearity and
characteristics of the nonlinear phenomena is little. In order to examine the characteristics
of nonlinearity in runoff phenomena we carried out numerical calculation using Eq.(5)
which is fundamental equation of runoff in this paper.
0
Runoff rate [mm/h]
30
–4/5
h
–1/5
25
]
β=0.8
q(0)=1.8[mm/h]
q(0)=1.4[mm/h]
q(0)=1.0[mm/h]
q(0)=0.6[mm/h]
q(0)=0.2[mm/h]
20
50
a0=0.2 [m
40
–4/5
h
–1/5
]
20
β=0.8
Runoff rate [mm/h]
a0=0.06 [m
0
r(t)=r0+r0sin(2πt–π/2)
Rainfall[mm/h]
r(t)=25+25sin(2πt–π/2)
50
Rainfall[mm/h]
40
q(0)=0.2[mm/h]
r0=20[mm/h]
r0=18[mm/h]
r0=16[mm/h]
r0=14[mm/h]
r0=12[mm/h]
r0=10[mm/h]
30
20
40
10
10
0
0
2
4
6
Time[h]
8
10
Figure 1. Relationship between initial
runoff rate and runoff rate
0
0
2
4
Time[h]
6
8
10
Figure 2. Relationship between rainfall
intensity and runoff rate
The nonlinearity of runoff phenomena about initial condition of soil moisture
content
The calculation is carried out by the change of initial runoff rate In order to examine the
effect of initial condition of soil moisture content. The calculation result is shown in
figure-1. It is proven that runoff rate increases nonlinearly without increasing in
proportion to linear increase of initial runoff rate, even if initial runoff rate simply
increased. And the hydrograph has been settled to the fixed curve with increase of initial
runoff rate and progress in the time. These facts will be able to be just called the
nonlinearity of runoff phenomena about initial condition of soil moisture content.
The nonlinearity of runoff phenomena about rainfall intensity
Rainfall intensity was changed in this calculation in order to examine the effect of rainfall
intensity about hydrograph. The calculation result is shown in figure-2. The peak value of
7
runoff rate becomes also the double, if rainfall intensity is made to be the double in linear
theory. However, in this calculation result, it does not become the simplicity for the
double, when rainfall intensity was made to be the double, and peak value of runoff rate
consists 3 times near. There is no linear relation for runoff rate and rainfall intensity.
These facts will be able to be just called the nonlinearity of runoff phenomena about
rainfall intensity.
The nonlinearity of runoff phenomena about the slope length
The change of hydrograph as slope length changes is shown in figure-3. What has been
required by this calculation is runoff rate [mm/h]. In linear theory, the slope length is not
influenced about runoff rate. The peak value of runoff rate decreases with the increase of
slope length and it converges on the fixed curve with the progress in the time. These facts
will be able to be just called the nonlinearity of runoff phenomena about the slope length.
The nonlinearity of runoff phenomena about the effective porosity
Finally, this calculation was carry out by the change of effective porosity for examining
the effect of the difference of spatial soil property of mountainous basins for hydrograph.
The result of this is shown in figure-4. It is proven that large difference occurs by the
change of effective porosity at hydrograph. The nonlinearity of runoff phenomena occurs
from the spatial distribution of soil property such as effective porosity in the basin.
It can be concluded that the nonlinearity of runoff phenomena remarkably arises
from the single slope.
：
：
：
：
：
：
10
0
0
2
4
6
Time[h]
L=10[m]
L=20[m]
L=30[m]
L=40[m]
L=50[m]
L=60[m]
8
20
40
10
20
Runoff rate [mm/h]
Runoff rate [mm/h]
β=0.8
q(0)=0.1[mm/h]
0
r(t)=20+20sin(2πt–π/2)
β=0.8
q(0)=0.1[mm/h]
： w=0.22
： w=0.32
： w=0.42
： w=0.52
20
40
Rainfall[mm/h]
20
30
Rainfall[mm/h]
0
r(t)=20+20sin(2πt–π/2)
10
0
0
Figure 3. Relationship between slope
length and runoff rate
2
4
Time[h]
6
8
10
Figure 4. Relationship between
effective porosity and runoff rate
ESTIMATION OF EFFECTIVE RAINFALL
Rainfall: r(t) in Eq.(5) which is fundamental equation for runoff proposed in this paper is
the effective rainfall. Though various estimation methods of effective rainfall are
8
500
： Observed
Loss of rainfall F(R) [mm]
： approximate curve
400
F=a*Tanh(b*R)
a=128.1
b=0.0054
300
200
100
0
0
100
200
300
400
500
Total rainfall R [mm]
Figure 5. Relationship between total
rainfall and loss of rainfall in the Kusaki
dam basin
The water–holding capacity distribution S(h)[1/mm]
proposed, there is no example of showing its validity. We propose two methods to
estimate effective rainfall in this section. One is to estimate effective rainfall from
observed data of discharge. Another method is to estimate effective rainfall by theory of
water holding capacity of a basin.
In this section, runoff parameters a0 and β is the average value which is simply
decided from observed recession curve of hydrograph using Eq.(22).
The application basin is Kusaki dam basin in Japan. Kusaki dam is located in the
Watarase river upstream 78km site of the Tone river water system. The drainage area of
Kusaki dam basin is 254km2, and it can be generally called a mountainous basin.
0.005
the water–holding capacity distribution：
2
3
S(h)=(1–AB)δ(h)+2AB sinh(Bh)/cosh (Bh)
0.004
0.003
a=128.1 b=0.00539
the proportion of the non–permeation area
part in a basin： 1–ab=0.31
0.002
0.001
0
0
100
200
300
400
500
600
The water–holding capacity h[mm]
Figure 6. The water-holding capacity
distribution in a Kusaki dam basin
The inverse estimation of effective rainfall from discharge
The effective rainfall is estimated from observation discharge data. When Eq.(5) which
is the fundamental equation is transformed, effective rainfall function is shown as follows,
r t   q t  
1
a 0 q t 

dq t 
dt
(27)
It is possible to require the effective rainfall by using this Eq.(27) from observation
discharge data.
The estimation method of effective rainfall by theory of water holding capacity in a
basin
Yamada proposed the theory of water holding capacity that is to obtain water-holding
capacity distribution from the proportion occupied in the basin of the water-holding
capacity of the soil as the estimation method of effective rainfall.
Rainfall does not contribute in the direct runoff, until it reaches the some values in
which accumulation amount of rainfall depends on the soil property in a basin. This time
9
accumulation amount of rainfall is defined as a water holding capacity. The water holding
capacity takes various values in actual basin. Then, it is defined as the proportion
occupied in the whole basin in the soil with any water holding capacity is the waterholding capacity distribution. The water holding capacity of the part with the basin is
made to be h, and the water holding capacity distribution is made to be S(h). Within total
rainfall R(t) by the time t, only the part which exceeded water holding capacity h flows. It
becomes the effective rainfall the result of integrating the product between the area
proportion S(h)dh and excess part by water holding capacity. Then, total loss F(R) is
expressed by Eq.(28). Eq.(28) is the first-kind Volterra type integral equation. It is
possible to obtain the solution by Laplace transformation as a Eq.(29).
R
F ( R)  R   ( R  h)S (h)dh
(28)
d 2 F  dF
S ( R)  
 1 
dR 2  dR
(29)
0

  ( R)
R 0 
where δ(R) is the Dirac's delta function, it means the proportion of the non-permeation
area part in the basin. Eq. (29) is used for the regression formula for the rerationship
Rainfall R(t) and Total loss F(R). where a and b are peculiar parameters in a basin.
Eq.(31) which shows Water-holding capacity distribution profile is obtained from Eq.(29)
and Eq.(30).
F ( R)  a tanh( bR )
(30)
sinh( bR)
S ( R)  (1  ab) ( R)  2ab
cosh 3 (bR)
2
(31)
The calculation result of total loss curve is shown in figure-5 and water-holding
capacity distribution is shown in figure-6. Effective rainfall obtained by the theory of
water holding capacity of a basin and the inverse estimation method is shown figure-7. In
comparison with the effective rainfall required from two different techniques, in the
initial stage, the effective rainfall required from the theory of water-holding capacity is
small, and it is proven to largely appear in the peak time. This reason is the theory of
water-holding capacity is obtained from the accumulation quantity of rainfall.
Finally, we carried out runoff analysis for Kusaki dam basin using two effective
rainfall that is obtained two theory proposed in this paper. The calculation result of runoff
analysis is shown in figure-8.
The results of runoff analysis match well with the observed data in Kusaki dam
basin. It can be concluded that the proposed lumped analysis method can express the
runoff in a mountainous basin effectively.
10
40
0
40
Rainfall[mm/h]
Total rainfall ： 266[mm]
： Effective rainfall
calculated by inverse estimation
： Effective rainfall
calculated by the theory of water–holding capacity
： observed
20
10
20
15
Runoff rate [mm/h]
Runoff rate [mm/h]
10
0
Rainfall[mm/h]
Runoff coefficient： 0.62
0
a0=0.040
β=0.91, m=10.1 20
30
： Calculated by inverse estimation
： Calculated by
the theory of water–holding capacity
10
●
Rainfall[mm/h]
1979.10.18 0:00～
20
： Observed
Initial runoff rate
： 0.07[mm/h]
5
0
Effective total rainfall
–20
0
Inverse estimation： 141.4[mm]
the theory of water–holding capacity ： 155.8[mm]
20
40
60
Time[h]
Figure 7. Calculated effective rainfall
by inverse estimation and the theory of
water-holding capacity in a Kusaki dam
basin
0
0
20
40
60
Time[h]
Figure 8. Calculated runoff rate using
effective rainfall in a Kusaki dam basin
CONCLUSION
1) The lumping runoff model proposed here can express the runoff in a mountainous
basin effectively.
2) Runoff parameters can be expressed by quantities of a surface soil layer, such as a
saturated hydraulic conductivity, an exponent in hydraulic conductivity function, the
thickness of surface soil layer and the effective porosity.
3) The nonlinearity in the runoff phenomena very greatly arises from the single slope.
REFERENCES
[1] Kure, S., Koshizuka Y. and Yamada, T., “Extraction of runoff characteristics from
flow recession characteristics of hydrograph”, In Japanese, Annual Journal of
Hydraulic Engineering, JSCE, Vol.48, (2004), pp 13-18.
[2] Yamada, T., “Studies on nonlinear runoff in mountainous basins”, In Japanese,
Annual Journal of Hydraulic Engineering, JSCE, Vol.47, (2003), pp259-264.
[3] Shimura, K., Yamada, T. and Ohara, N. and Matsuki,H., “Study on runoff
characteristics of the large-scale channel network using a physically based model ”,
In Japanese, Journal of Japan society of hydrology and water resources ,JSCE,
Vol.14, (2001), pp217-228